On a nonlinearly Dirichlet problem with a singularity along the boundary

Abstract

Let $\Omega$ be a $C^{1,1}$ and bounded domain in $R^{n},$ with $n\geq2.$ Let $K$ and $f$ be two nonnegative functions on $\Omega$ belonging to $L^{p} ( \Omega ) $ for some $p>n/2$ and let $\alpha>0.$ In this paper we prove existence and uniqueness of a strong solution $u\in W_{loc}^{2,p} ( \Omega ) \cap C ( \overline{\Omega} ) $ for the elliptic Dirichlet problem $-\Delta u+\lambda u=Ku^{-\alpha}+f$ in $\Omega,$ $u>0$ in $\Omega,$ $u=0$ in $\partial\Omega.$ Moreover, for the case $\lambda=0,$ $f=0,$ we prove, under the weaker hypothesis $p>\frac{ ( \alpha^{2}+1 ) n}{2\alpha^{2}+n},$ the existence of a solution $u\in W_{loc}^{2,p} ( \Omega ) $ for the above problem that satisfies, in a suitable extended sense, the boundary condition $u=0$ on $\partial\Omega.$